Khovanov homology is a categorification of the Jones polynomial.
It is known that Khovanov homology also arises from a categorical representation of braid groups, so we can regard it as a kind of quantum knot invariant.
However, in contrast to the case of classical quantum invariants, its relation to Vassiliev invariants remains unclear.
Aiming at the problem, in this talk, we discuss a categorified version of Vassiliev skein relation on Khovanov homology. The idea is that we realize the "wall-crossing" in the embedding space as a morphism between Khovanov complexes. It will be seen that the homotopy type of its mapping cone defines a singular link invariant, which can be thought of as a "derivative" of Khovanov homology in view of Vassiliev theory. Furthermore, we will compute first derivatives to determine Khovanov homologies of twist knots.

This talk is based on papers arXiv:2005.12664 (joint work with N.Ito) and arXiv:2007.15867.

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